Soliton resolution for the complex short pulse equation with weighted Sobolev initial data (2101.12697v1)

Abstract: We employ the $\bar{\partial}$-steepest descent method in order to investigate the Cauchy problem of the complex short pulse (CSP) equation with initial conditions in weighted Sobolev space $H^{1,1}(\mathbb{R})={f\in L^{2}(\mathbb{R}): f',xf\in L^{2}(\mathbb{R})}$. The long time asymptotic behavior of the solution $u(x,t)$ is derived in a fixed space-time cone $S(x_{1},x_{2},v_{1},v_{2})={(x,t)\in\mathbb{R}^{2}: y=y_{0}+vt, ~y_{0}\in[y_{1},y_{2}], ~v\in[v_{1},v_{2}]}$. Based on the resulting asymptotic behavior, we prove the solution resolution conjecture of the CSP equation which includes the soliton term confirmed by $N(I)$-soliton on discrete spectrum and the $t^{-\frac{1}{2}}$ order term on continuous spectrum with residual error up to $O(t^{-1})$.